Nonperturbative Green's function method to determine the electronic spectral function due to electron-phonon interactions: Application to a graphene model from weak to strong coupling

In solid state physics, the electron-phonon interaction (EPI) is central to many phenomena. The theory of the renormalization of electronic properties due to EPIs became well established with the theory of Allen-Heine-Cardona (AHC), which is usually applied to second order in perturbation theory. However, this is only valid in the weak coupling regime, while strong EPIs have been reported in many materials. As a result, and with AHC becoming more established through density-functional perturbation theory, some nonperturbative (NP) methods have started to arise in the last years. However, they are usually not well justified and it is not clear to what degree they reproduce the exact theory. To address this issue, we present a stochastic approach for the evaluation of the nonperturbative interacting Green's function in the adiabatic limit, and show it is equivalent to the Feynman expansion to all orders in the perturbation. Also, by defining a self-energy, we can reduce the effect of broadening needed in numerical calculations, improving convergence in the supercell size. In addition, we clarify whether it is better to average the Green's function or self-energy. Then we apply the method to a graphene tight-binding model, and we obtain several interesting results: (i) The Debye-Waller term, which is normally neglected, does affect the change of the Fermi velocity vF, and should be included to obtain accurate results. (ii) Although at room temperature second-order perturbation theory (P2) agrees well with the NP change of vF and of the self-energy close to the Dirac point, at high temperatures there are significant differences. For other k points, the disagreement between the P2 and NP self-energies is visible even at low temperatures, raising the question of how well P2 works in other materials. (iii) Close enough to the Dirac point, positive- and negative-energy peaks merge, giving rise to a single peak. (iv) At strong coupling and high temperatures, a peak appears at ω=0 for several states, which is consistent with previous works on disorder and localization in graphene. (v) The spectral function becomes more asymmetric at stronger coupling and higher temperatures. Finally, we show that the method has better convergence properties when the coupling is strong relative to when it is weak, and discuss other technical aspects.